Asymptotic zeros of Poincaré series

Abstract

We study the zeros of Poincaré series $P_{k,m}$ for the full modular group. We consider the case where $m\sim \alpha k$ for some constant $\alpha >0$. We show that in this case a positive proportion of the zeros lie on the line $\frac{1}{2}+it$. We further show that if $\alpha >\frac{\log (2)}{2\pi }$ then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when $\alpha$ is small.